We read the problem together and decided to try a number to see what would happen.
4 lit up only the yellow light, so we made a list of the rules it could be - even numbers, numbers in the 2x table, numbers in the 4x table or numbers less than 10.
We decided to continue finding out what rule yellow could be, so we next tried 6 as it was an even number to see if the 2x table rule could be true. Yellow light was the only one lit up again. So now we know it cant be numbers in the 4x table. So then we tired 12 to check the even number rule - yellow and red lit up. So we thought yellow must be even numbers. And we made a list of rules for red as this was the first time it had come on - numbers more than 10, two digit numbers, numbers in the 12x table.
We thought we would check to see if the rule for red was numbers 10 or more so we tried 10 - but this time yellow, red and blue lit up. So this meant that the rule couldn't be numbers 10 or more but it could be 2 digit numbers. We guessed that blue could be numbers in the 10x table as this is the first time it had lit up. To test if blue came on for 10x and red was 2 digits we tried 100. Only blue came on. This would fit the rule red is 2 digit numbers and blue is 10x tables.
So we looked at the yellow rule again. We had said it was even numbers, so to test this we realised it should stay off for an odd number and we hadn't tried any odd numbers yet. We chose to use 9 - but yellow was still lit! And green came on too. This was the first time green had come on and the first time we had used an odd number so we thought the rule for green could be odd numbers. To try this idea, we used 15 - all the lights lit up. So now we realised blue cant be 10x but could be 5x.
At this point we were loosing track of which numbers met each rule and we couldn't see a pattern, so Mrs Green suggested we make a list for each light (see attached). This way we could test the rule that we had already thought of for each colour.
Under each colour we wrote the numbers we had already tested to look for a pattern - blue was the most obvious so we checked that first and it lit up for 5 and 500 so we thought that we had tried enough numbers to prove the rule 'When the number is from the 5x table the blue light will stay on'.
Next we looked at the pattern of numbers for green - they were all odd numbers, so we tried a few random even numbers to prove the rule. The green light didn't stay on for the even numbers so now we had enough evidence to prove the rule 'When the number is odd the green light will stay on'.
Then we were left with yellow and red. We could not see an obvious pattern, so we started to write down numbers we tried that made it stay off. For yellow we noticed any large numbers we tried it hadn't come on, so thought it could be a number below 100. We tried 50, and it didn't come on. So then we thought the rule would be numbers less than 50, but when we tired 44 none of the lights came on. So we kept trying each number in order from 44 down until the yellow light came back on - it stayed on when we got to 30. We had already tried numbers less that 30 so now we had enough to prove that the rule must be 'When the number is 30 or less the yellow light stays on'.
So then we were left with the red rule. We looked at the numbers we had tried already to think of a rule. Looking at the numbers we had already tried we thought maybe it could be 2 digit numbers - but then we remembered that when we tried 44 none of the lights came on. But we tried 100 to see what would happen, and the red light stayed off, but then we tried 775 and it stayed on. So again we started to write a list of numbers it stayed off for and we noticed that the tens digit was even when it turned off. We tried 98 and it stayed on. Looking over the numbers we think the final rule is 'When the number has an odd number of tens the red light stays on'.
So now we know the rules, we can work out the lowest number that will turn them all on. To turn red on, it must be an odd number of tens, which means it has to be a two digit number. It also has to be in the 5x table to turn on blue - 10, 15, 20, 25, 30, 35... But it cant be 35 or yellow will go off. 10, 15, 30 have an odd number of tens for red, and in the 5x table for blue and are under 30 for yellow. It has to be an odd number for green so the only number it could be is 15. When we tried 15, all the lights came on.